Study establishes monodromy equivalence for Lamé-type equations and constructs cone spherical metrics.
problem Investigating monodromy equivalence and finite-gap structures of Lamé-type equations.
method Analyzing finite-gap structures and constructing cone spherical metrics.
result Established monodromy equivalence between classical and generalized Lamé-type equations, derived finite-gap structures, and constructed cone spherical metrics.
Paper provides an alternative proof for Dahmen's conjectures about Lame equations.
problem Proving the conjecture about the number of Lame equations with finite monodromy.
method Applying results on spherical tori with one conical singularity to Lame equations.
result Alternative proof of Dahmen's conjectures.
Some special solutions to the multidimensional Lamé and Bourlet type equations are constructed in an explicit form.
Extremal spectral properties of Lawson tau-surfaces are investigated. The Lawson tau-surfaces form a two-parametric family of tori or Klein bottles minimally immersed in the standard unitary three-dimensional sphere. A Lawson tau-surface carries an extremal metric for some eigenvalue of the Laplace-Beltrami operator. U…
Characterizes Lamé equations with finite monodromy on flat tori.
problem Classifying Lamé equations with finite monodromy on flat tori.
method Combining dessin d'enfants with geometry of spherical tori to prove existence and provide descriptions.
result Finiteness of (B,τ) for given (n,M) with not∈frac12+Z and explicit counting formula. We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé's system of equations. We show that the symmetry group of the Lamé's system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dep…
Integrable flows on null curves in anti-de Sitter 3-space studied.
problem Analyzing integrable flows on null curves in anti-de Sitter 3-space.
method Formulated integrable flows related to the KdV hierarchy on null curves exploiting the geometry of anti-de Sitter 3-space.
result Explicitly found closed stationary solutions in terms of periodic solutions of a Lamé equation.
Abstract: Determines Lamé coefficients from boundary measurements.
problem Determining Lamé coefficients from elastic boundary measurements.
method Explicit symbol of elastic Dirichlet-to-Neumann map, partial derivatives determination.
result Elastic Dirichlet-to-Neumann map uniquely determines Lamé coefficients.
We discuss a Lie algebraic and differential geometry construction of solutions to some multidimensional nonlinear integrable systems describing diagonal metrics on Riemannian manifolds, in particular those of zero and constant curvature. Here some special solutions to the Lamé and Bourlet type equations, determining by…
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
We show how Ramond free neutral Fermi fields lead to a τ-function theory of BKP type which describes iso-orthogonal deformations of systems of ortogonal curvilinear coordinates. We also provide a vertex operator representation for the classical Ribaucour transformation.
The vectorial fundamental transformation for the Darboux equations is reduced to the symmetric case. This is combined with the orthogonal reduction of Lame type to obtain those vectorial Ribaucour transformations which preserve the Egoroff reduction. We also show that a permutability property holds for all these transf…
Study calculates geometric invariants from Navier-Lamé spectrum.
problem Determine geometric properties from elastic spectrum.
method Explicitly calculates coefficients of asymptotic expansion of spectrum.
result An n-dimensional ball is uniquely determined by its Navier-Lamé spectrum. We solve the problem of description for nonsingular pairs of compatible flat metrics in the general N-component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. Th…
Geometrically revisits Dupin cyclidic systems using evolving circles and cyclides.
problem Understanding the geometric properties and evolution of Dupin cyclidic systems.
method Evolving initial circles or Dupin cyclides to generate Lamé families of Dupin cyclidic systems in various space forms.
result Lamé families are parallel surfaces in different space forms.
The space of Lamé functions is mapped to a Riemann surface with known topology.
problem Understanding the structure of the space of Lamé functions and its relation to Abelian integrals.
method Isomorphic mapping to elliptic curves and Abelian differentials with specific properties.
result The space of Lamé functions is a Riemann surface of finite type with known genus and Euler characteristic.
We deal with the problem of description of nonsingular pairs of compatible flat metrics for the general N-component case. We describe the scheme of the integrating the nonlinear equations describing nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics). It is based on …
For a bounded domain Ω⊂Rn with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the trace of the strongly continuous semigroup associated with the Navier-Lamé operator on Ω as t→0+. These coefficients (i.e., spectral invariants) provide precise …
Study of symmetries in a 2D space with specific metric properties.
problem Understanding symmetries in a 2D space with diagonal metrics.
method Analyzing Killing vector fields under specific restrictions on Lamé coefficients.
result Concretely described symmetries of the metric under given conditions.
The nonlinear equations for the general nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of inverse scattering problem is proved. For these equations, the Lax pairs with a spectral parameter are presented. Moreover, we demo…
The paper provides estimates for eigenvalues of elliptic differential problems.
problem Computing eigenvalue estimates for elliptic differential problems.
method Analytical computation of eigenvalues for specific types of elliptic differential equations.
result Universal estimates of eigenvalues and gaps between consecutive eigenvalues are derived.
A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere S3=R3∪{∞}. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type…
The paper studies self-Bäcklund curves in centroaffine geometry using elliptic functions.
problem Understanding self-Bäcklund curves in centroaffine geometry.
method Description of general properties and detailed analysis using elliptic functions.
result Provides a detailed description of self-Bäcklund centroaffine curves in terms of elliptic functions.
We investigate reductions of the two-dimensional Dirac equation imposed by the requirement of the existence of a differential operator Dn of order n mapping its eigenfunctions to adjoint eigenfunctions. For first order operators these reductions (and multi-component analogs thereof) lead to the Lame equations desc…
We discuss the problem of R-separability (separability of variables with a factor R) in the stationary Schrödinger equation on n-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of R-separability in PDE (Laplace equation on E3…
Sequences of Levy transformations for the Darboux system of conjugates nets in multidimensions are studied. We show that after a suitable number of Levy transformations, with at least a Levy transformation in each direction, we get closed formulae in terms of multi-Wroński determinants. These formulae are for the tange…
Low redispatch prices boost green hydrogen production cost, encouraging electrolyzer siting.
problem Uncertainty in redispatch power availability and its impact on green hydrogen production cost.
method Historic redispatch time series analysis and power purchase scenarios evaluation.
result Low price levels can lead to notable production cost reductions, incentivizing electrolyzer siting.
Study Galois groupoids of discret Painlevé equations.
problem Computing Galois groupoids for discret Painlevé equations.
method Using semi-continuity theorem for Galois groupoid in confluence of difference to differential equations.
result Computed Galois groupoids for discret Painlevé equations.
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
We present an unsupervised approach for discovering semantic representations of mathematical equations. Equations are challenging to analyze because each is unique, or nearly unique. Our method, which we call equation embeddings, finds good representations of equations by using the representations of their surrounding …
Paper establishes estimates for nonlinear equations on compact manifolds.
problem Estimating solutions to fully nonlinear equations with gradient terms on compact almost Hermitian manifolds.
method Establishes second order estimates and proves existence of solutions for specific equations.
result Proves existence of solutions for various equations, including Monge-Ampère and Hessian equations.
Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
problem Elliptic equations on hypercomplex manifolds.
method Proves C^2,alpha estimates under suitable assumptions.
result Solutions to specific elliptic equations on hyperkähler manifolds satisfy C^2,alpha estimates.
The paper generalizes Monge-Ampère equations and their solutions in differential geometry.
problem Understanding the structure of Monge-Ampère equations and their solutions.
method Generalizing Monge-Ampère equations to higher-order systems and proving their solutions correspond to integral manifolds of exterior differential systems.
result The Korteweg-de Vries (KdV) equation and Cauchy-Riemann equations are examples of generalized Monge-Ampère equations.
We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…
Introduces a new PDE involving differential forms for Kähler geometry.
problem Solving a unified PDE for various important equations in Kähler geometry.
method Introduces a fully nonlinear PDE with differential form Λ and proves solvability conditions.
result Generalizes previous works and proves a conjecture for the dHYM equation.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in S1,} \end{equation} where (−Δ)21 stands for the fractional Laplacian and κ is a bounded function. We interpret the above equation as the prescri…
The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
Paper solves Hessian equations on Kähler manifolds.
problem Solving Hessian equations on Kähler manifolds.
method Combines elementary symmetric functions; provides sufficient and necessary condition.
result Generalizes results for Hessian and Hessian quotient equations.
The paper introduces new equations in Kähler geometry and proves their solutions and convexity.
problem Solving equations in Kähler geometry and understanding their geometric implications.
method Using moment map pictures to motivate and prove solutions for the equations.
result The Mabuchi functional for certain equations is shown to be convex.
We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…
Probabilistic grammars improve equation discovery from data.
problem Discovering scientific laws from data using equations.
method Proposed probabilistic context-free grammars to encode soft constraints and a Monte-Carlo algorithm.
result Probabilistic grammars lead to more efficient equation discovery.
In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equa…
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equ…
In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averag…
Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.
problem Deriving new equations for magnetic systems and proving their well-posedness.
method Introducing the magnetic Euler-Arnold equation and proving well-posedness for specific equations.
result Local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.